Nonorthogonality relations between complex hybrid modes: an application for the leaky-wave analysis of laterally shielded top-open planar transmission lines

In this paper, the orthogonality relations between complex modes are investigated to demonstrate the coupling effect that exists along a transverse direction (z-axis) between TE^z and TM^z hybrid modes when power storage, loss, or leakage occur in the longitudinal direction (y-axis) of a transmission line. As an example, the parallel-plate waveguide modes are used to study a laterally shielded top-open microstrip line. The study of leaky waves is a very interesting application of the novel orthogonality relations derived in this paper. This study is carried out using the correct and incorrect orthogonality formulations, showing that physically valid results are obtained only when the orthogonality relations derived in this paper are properly introduced in the formulation. A physical explanation of this novel coupling effect between TE^z and TM^z hybrid modes with respect to the y-direction is given in terms of inhomogeneous plane waves. Comparisons with previous results for a laterally shielded slot-line antenna are presented to confirm the accuracy and usefulness of this novel proposed method.